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Published: 29.05.2021  In probability theory , a normal or Gaussian or Gauss or Laplace—Gauss distribution is a type of continuous probability distribution for a real-valued random variable.

Distribution Theory

A physical variable is customarily thought of as a function, i. For example, if the independent variable is time t and the physical quantity is a force f , then one would say that the force is known if its value f t is specified at every instant of time t.

However, it is impossible to observe the instantaneous values of f t. Any measuring instrument would merely record the effect that f produces on it over some nonvanishing interval of time. As we shall see, another way of describing a physical variable is to specify it as a functional, i. We shall be exclusively concerned with functionals of a special type, namely, distributions.

It turns out that the distribution concept provides a better mechanism for analyzing certain physical phenomena than does the function concept because, for one reason, various entities, such as the delta function, which arise naturally in several mathematical sciences can be correctly described as distributions but not as functions.

Moreover, any physical quantity that can be adequately represented as a function can also be characterized as a distribution and, indeed, there is an advantage in using the latter representation. One cannot assign instantaneous values to a distribution, and consequently the problem of physically interpreting such values does not arise.

Actually, the idea of specifying a function not by its values but by its behavior as a functional on some space of testing functions is a concept that is quite familiar to scientists and engineers through their experience with the classical Fourier and Laplace transformations. When specifying a function f t by its Fourier transform. On the other hand, the testing functions on which distributions are defined cannot, in general, be written down in an explicit form. Nevertheless, the functional concept is a basic one both for Fourier transforms and for distributions.

We shall now take up in this first chapter the definitions of testing functions and distributions, their basic properties, and certain operations that one can apply to them. Before we can describe distributions, we must define the testing functions, on which distributions operate. Several different spaces of testing functions will be discussed in this book. The one that is considered in this section is, for our purposes, the most important and is the one we shall employ most often.

Throughout this and the next section, the independent real variable t will be assumed to be one-dimensional. When a function has continuous derivatives of all orders on some set of points, we shall say that the function is infinitely smooth on that set. If this is true for all points, we shall merely say that the function is infinitely smooth.

Moreover, whenever we refer to a complex number or a complex-valued function, it is understood that the number may be real or the function may be real-valued. The reader can find in Appendix A the axioms that a space must satisfy if it is to be a linear space obviously satisfies them. It is a fact that any complex-valued function f t that is continuous for all t and zero outside a finite interval can be approximated uniformly by testing functions. Hence, we may differentiate 4 under the integral sign with respect to t Titchmarsh , p.

Moreover, we have already shown [in our discussion of. In addition, it is zero outside some finite interval. This is what we wished to show. The numbers Nk may vary with k. Throughout this book we shall associate with each new space of functions or distributions no more than one concept of convergence.

By saying that such a space is closed under convergence, we shall mean that all its convergent sequences have limits that are also in the space. Actually, this property of being closed under convergence is related to the completeness of these spaces. In order to define completeness properly, one must introduce the concepts of Cauehy sequences and neighborhoods for these spaces.

In this book, we are going to avoid such discussions, and we shall therefore refrain from using the word complete. In this regard, see Prob. We shall usually deal with convergent sequences, but we shall also consider convergent directed sets occasionally. In other words, it is E plus all the limit points of E. A functional. Distributions, that possess, in addition, two essential properties. The first of these is linearity.

The second property is continuity. If f is known to be linear, the definition of continuity may be somewhat simplified. In this case, f to zero. A continuous linear functional on the space is a distribution.

One way to generate distributions is as follows. Let f t be a locally integrable function i. Corresponding to f t , we can define a distribution f through the convergent integral.

Next, we observe that. Upon substituting this inequality into the right-hand side of 3 and noting that. Distributions that can be generated through 2 from locally integrable functions are called regular distributions. It is a fact that two continuous functions that produce the same regular distribution are identical. On the other hand, if f t is merely a locally integrable function, the regular distribution corresponding to it certainly determines the function f t uniquely within every open interval where it is known to be continuous.

This follows directly from the argument given above. However, the regular distribution does not determine f t at points where f t is discontinuous. In fact, we may alter the values of f t on a set of measure zero without altering the regular distribution. The positive n th root is understood.

Indeed, we have. Thus, we have shown that a regular distribution determines the function producing it almost everywhere. Actually, the values of a function on a set of measure zero are unimportant so far as Lebesgue integration is concerned, and this attitude is also usually adopted in physical applications. All locally integrable functions that differ at most on a set of measure zero determine an equivalence class of functions, and these functions produce the same regular distribution.

Hence, without ambiguity we may consider an equivalence class of functions and its regular distribution as being the same entity. We shall at times use the same expression to denote a regular distribution and a function that generates it. An even more ambiguous situation can arise when the function is constant. For instance, the symbol c can represent the number c, the function that equals c for all values of its argument, or the corresponding regular distribution.

For the sake of brevity, we shall not always specify explicitly which of these meanings such a symbol has. The context in which it is used should clarify any possible ambiguity. In certain parts of modern analysis, such as in functional analysis, it has become essential to distinguish between the value of a function at a point t and the function itself.

In these cases, it is conventional to let f t be the numerical value corresponding to the value t and to let f denote the function i. This is in contrast to classical analysis, where such a distinction was not needed. Nor will our symbolism make this distinction. In general, both f t and f will represent a regular distribution or a function. However, at times the symbols f 0 , f T , f c , etc. Again, our intent should be discernible from the context in which these symbols are used.

The importance of the class of distributions stems from the fact that not only does it include representations of locally integrable functions i.

Moreover, many operations, such as integration, differentiation, and other limiting processes that were originally developed for functions, can be extended to these new entities. It should be mentioned, however, that other operations such as the multiplication of functions f t g t or the formation of composite functions f g t cannot be extended, in general, to all distributions. This constitutes a disadvantage of distribution theory. However, this distribution cannot be obtained from a locally integrable function through the use of 2.

This would be a contradiction. Actually, its name delta function is a misnomer and, for this reason, we shall refer to it as the delta functional.

Before leaving this section, let us say a few more words about notation. If f of testing functions, where the independent variable is t, f will also be designated by f t , the symbol heretofore used for regular distributions or ordinary functions.

As was indicated before, when f is a regular distribution, this alternative notation does not lead to any essential ambiguity in view of the one-to-one relation between an equivalence class of functions composed of all functions differing only on a set of measure zero and the corresponding regular distribution.

On the other hand, if f is a singular distribution, the notation f t is merely a convenient symbolism. It has the advantage of displaying the variable t that occurs in the testing functions. This will be useful when we define for distributions such operations as convolution, Fourier and Laplace transformation, and certain changes of variable. Henceforth, when we designate a singular distribution by f t , it will be understood that this is not a function or a regular distribution.

Friedman , refer to f t in this case as a symbolic function. Let us give an illustration. One often finds in the literature the expression. It is understood that the right-hand side of 9 has no meaning other than that given to it by the left-hand side. The intervening formal manipulations in 10 merely demonstrate a consistency with what we would do if we were dealing with functions.

Example 1. As we have already stated, the delta functional is not a function and therefore cannot be plotted. There is value in this sort of intuitive thinking. It not only gives one a better grasp of the concept of the delta functional but also provides a method for discovering other singular distributions, as we shall see. Nevertheless, we must not ignore the fact that these considerations are not rigorous; indeed, all our conjectures will be proved subsequently in terms of distribution theory.

As was shown in that section, for every continuous function f the integral. Actually, the form of the approximating pulse is not important so long as it is nonnegative and has a unit area concentrated mainly over a very small interval around the origin. In Sec. Distribution Theory Distribution Theory

Skip to search form Skip to main content You are currently offline. Some features of the site may not work correctly. We introduce the theory of distributions and examine their relation to the Fourier transform. We then use this machinery to find solutions to linear partial differential equations, in particular, fundamental solutions to partial differential operators. Finally, we develop Sobolev spaces in order to study the relationship between the regularity of a partial differential equation and its solution, namely elliptic regularity.

In probability theory and statistics , the characteristic function of any real-valued random variable completely defines its probability distribution. If a random variable admits a probability density function , then the characteristic function is the Fourier transform of the probability density function. Thus it provides an alternative route to analytical results compared with working directly with probability density functions or cumulative distribution functions. There are particularly simple results for the characteristic functions of distributions defined by the weighted sums of random variables.

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